3.75 \(\int x^3 \sqrt{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=95 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{64 a^4}-\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)} \]

[Out]

(-3*Sqrt[ArcCos[a*x]])/(32*a^4) + (x^4*Sqrt[ArcCos[a*x]])/4 - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*
x]]])/(64*a^4) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(16*a^4)

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Rubi [A]  time = 0.194688, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4630, 4724, 3312, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{64 a^4}-\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[ArcCos[a*x]],x]

[Out]

(-3*Sqrt[ArcCos[a*x]])/(32*a^4) + (x^4*Sqrt[ArcCos[a*x]])/4 - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*
x]]])/(64*a^4) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(16*a^4)

Rule 4630

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCos[c*x])^n)/(m
+ 1), x] + Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int x^3 \sqrt{\cos ^{-1}(a x)} \, dx &=\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}+\frac{1}{8} a \int \frac{x^4}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{32 a^4}-\frac{\operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}\\ &=-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{64 a^4}-\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}\\ \end{align*}

Mathematica [C]  time = 0.109826, size = 125, normalized size = 1.32 \[ -\frac{\sqrt{\cos ^{-1}(a x)} \left (4 i \sqrt{2} \cos ^{-1}(a x) \text{Gamma}\left (\frac{3}{2},-2 i \cos ^{-1}(a x)\right )+4 \sqrt{2} \sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{3}{2},2 i \cos ^{-1}(a x)\right )+i \cos ^{-1}(a x) \text{Gamma}\left (\frac{3}{2},-4 i \cos ^{-1}(a x)\right )+\sqrt{\cos ^{-1}(a x)^2} \text{Gamma}\left (\frac{3}{2},4 i \cos ^{-1}(a x)\right )\right )}{128 a^4 \left (-i \cos ^{-1}(a x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*Sqrt[ArcCos[a*x]],x]

[Out]

-(Sqrt[ArcCos[a*x]]*((4*I)*Sqrt[2]*ArcCos[a*x]*Gamma[3/2, (-2*I)*ArcCos[a*x]] + 4*Sqrt[2]*Sqrt[ArcCos[a*x]^2]*
Gamma[3/2, (2*I)*ArcCos[a*x]] + I*ArcCos[a*x]*Gamma[3/2, (-4*I)*ArcCos[a*x]] + Sqrt[ArcCos[a*x]^2]*Gamma[3/2,
(4*I)*ArcCos[a*x]]))/(128*a^4*((-I)*ArcCos[a*x])^(3/2))

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Maple [A]  time = 0.08, size = 91, normalized size = 1. \begin{align*}{\frac{1}{128\,{a}^{4}} \left ( -\sqrt{2}\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -8\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +16\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) +4\,\arccos \left ( ax \right ) \cos \left ( 4\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(a*x)^(1/2),x)

[Out]

1/128/a^4/arccos(a*x)^(1/2)*(-2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)
)-8*Pi^(1/2)*arccos(a*x)^(1/2)*FresnelC(2*arccos(a*x)^(1/2)/Pi^(1/2))+16*arccos(a*x)*cos(2*arccos(a*x))+4*arcc
os(a*x)*cos(4*arccos(a*x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acos(a*x)**(1/2),x)

[Out]

Integral(x**3*sqrt(acos(a*x)), x)

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Giac [B]  time = 1.28833, size = 255, normalized size = 2.68 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\sqrt{2}{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{256 \, a^{4}{\left (i - 1\right )}} + \frac{\sqrt{\pi } i \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (4 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-4 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2}{\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{256 \, a^{4}{\left (i - 1\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^(1/2),x, algorithm="giac")

[Out]

1/256*sqrt(2)*sqrt(pi)*i*erf(-sqrt(2)*(i + 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) + 1/32*sqrt(pi)*i*erf(-(i + 1)*
sqrt(arccos(a*x)))/(a^4*(i - 1)) + 1/64*sqrt(arccos(a*x))*e^(4*i*arccos(a*x))/a^4 + 1/16*sqrt(arccos(a*x))*e^(
2*i*arccos(a*x))/a^4 + 1/16*sqrt(arccos(a*x))*e^(-2*i*arccos(a*x))/a^4 + 1/64*sqrt(arccos(a*x))*e^(-4*i*arccos
(a*x))/a^4 - 1/256*sqrt(2)*sqrt(pi)*erf(sqrt(2)*(i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1)) - 1/32*sqrt(pi)*erf((
i - 1)*sqrt(arccos(a*x)))/(a^4*(i - 1))